![]() Results using the haversine formula may have an error of up to 0.5% because the Earth is not a perfect sphere, but an ellipsoid with a radius of 6,378 km (3,963 mi) at the equator and a radius of 6,357 km (3,950 mi) at a pole. The great-circle distance is the shortest distance between two points along the surface of a sphere. It is formed by the intersection of a plane and the sphere through the center point of the sphere. A great circle (also orthodrome) of a sphere is the largest circle that can be drawn on any given sphere. The haversine formula works by finding the great-circle distance between points of latitude and longitude on a sphere, which can be used to approximate distance on the Earth (since it is mostly spherical). In the haversine formula, d is the distance between two points along a great circle, r is the radius of the sphere, ϕ 1 and ϕ 2 are the latitudes of the two points, and λ 1 and λ 2 are the longitudes of the two points, all in radians. The haversine formula can be used to find the distance between two points on a sphere given their latitude and longitude: There are a number of ways to find the distance between two points along the Earth's surface. Given the two points (1, 3, 7) and (2, 4, 8), the distance between the points can be found as follows: d =ĭistance between two points on Earth's surface Like the 2D version of the formula, it does not matter which of two points is designated (x 1, y 1, z 1) or (x 2, y 2, z 2), as long as the corresponding points are used in the formula. Where (x 1, y 1, z 1) and (x 2, y 2, z 2) are the 3D coordinates of the two points involved. The distance between two points on a 3D coordinate plane can be found using the following distance formulaĭ = √ (x 2 - x 1) 2 + (y 2 - y 1) 2 + (z 2 - z 1) 2 For example, given the two points (1, 5) and (3, 2), either 3 or 1 could be designated as x 1 or x 2 as long as the corresponding y-values are used: The order of the points does not matter for the formula as long as the points chosen are consistent. Where (x 1, y 1) and (x 2, y 2) are the coordinates of the two points involved. MathWorld-A Wolfram Web Resource.The distance between two points on a 2D coordinate plane can be found using the following distance formula Referenced on Wolfram|Alpha Sphere Packing Cite this as: Penguin Dictionary of Curious and Interesting Geometry. ![]() Penguin Dictionary of Curious and Interesting Numbers. "Is Random Close Packing of Spheres Well Defined?" ![]() Oxford, England: Oxford University Press,Įrror-Correcting Codes Through Sphere Packings to Simple Groups. On-Line Encyclopedia of Integer Sequences." Steinhaus, H. "On the Densest Packing of Spheres in a Cube." Can. Cambridge, England: Cambridge University Press, 1964. "Putting the Best Face of a Voronoi Polyhedron." Proc. "Physics of Granular States." Science 255, "On the Densest Packing of Equal Spheres in a Cube." Math. "Dense Packings of Equal Spheres in a Cube." Electronic J. "Besprechung des Buchs von L. A. Seeber: Intersuchungen überĭie Eigenschaften der positiven ternären quadratischen Formen usw." Göttingsche "Packing Spheres."Ĭolossal Book of Mathematics: Classic Puzzles, Paradoxes, and Problems. Gardner's New Mathematical Diversions from Scientific American. In der Ebene, auf der Kugel und in Raum, 2nd ed. ![]() "The Problem of Packing a Number of Equal NonoverlappingĬircles on a Sphere." Trans. "Close-Packing and so Forth." Illinois J. "Probable Nature of the Internal Symmetry of Crystals." Nature 29, 186-188, 1883. The results of Gensane (2004) improve those Compressing a random packing gives polyhedra with an averageįor sphere packing inside a cube, see Goldberg (1971), Schaer (1966), Gensane (2004), and Friedman. Random close packing of spheres in three dimensions gives packing densities in the range 0.06 to 0.65 (Jaeger and Nagel 1992, TorquatoĮt al. This is the lattice formed by carbonĪtoms in a diamond (Conway and Sloane 1993, p. 113). Hilbert and Cohn-Vossen (1999, pp. 48-50) consider a tetrahedral lattice packing in which each sphere touches four neighbors and the density is. Must touch at least four others, and the four contact points cannot be in a single Reported by Hilbert and Cohn-Vossen (1999, p. 51). The rigid packing with lowest density known has (Gardner 1966), significantly lower than that
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